Noname manuscript No. (will be inserted by the editor) Sublogarithmic Distributed MIS Algorithm for Sparse Graphs using Nash-Williams Decomposition Leonid Barenboim ¤ ¢ Michael Elkin ¤ Received: date / Accepted: date Abstract We study the distributed maximal indepen- 1 Introduction dent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a subloga- 1.1 Distributed Message Passing Model rithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the ¯rst We study symmetry breaking problems in computer sublogarithmic algorithm for computing MIS on graphs networks. The network is modeled by an undirected un- of bounded arboricity. This is a large family of graphs weighted n-vertex graph G = (V; E). The processors in that includes graphs of bounded degree, planar graphs, the network are represented by the vertices of G. For graphs of bounded genus, graphs of bounded treewidth, each two vertices u; v 2 V , there is an edge (u; v) 2 E if graphs that exclude a ¯xed minor, and many other and only if the two processors corresponding to u and v graphs. We also devise e±cient algorithms for coloring in the network are connected by a communication link. graphs from these families. The processors communicate over the edges of G. These results are achieved by the following tech- Traditionally, symmetry breaking problems have nique that may be of independent interest. Our algo- been studied in the synchronous model [5,19,11,18]. rithm starts with computing a certain graph-theoretic In this model the communication proceeds in discrete structure, called Nash-Williams forests-decomposition. rounds. There is a global clock that is accessible to all Then this structure is used to compute the MIS or col- the vertices which counts the rounds. In each commu- oring. Our results demonstrate that this methodology nication round each vertex v 2 V can send a short is very powerful. message, of size O(log n) bits, to each of its neighbors, and these messages arrive before the next round starts. Finally, we show nearly-tight lower bounds on the In addition, it can perform local computations based on running time of any distributed algorithm for comput- the information from messages that it has recieved so ing a forests-decomposition. far. For an algorithm A in this model, the running time of A is the (worst-case) number of rounds of distributed communication that may occur during an execution of Keywords MIS ¢ Coloring ¢ Arboricity ¢ Forests- A. Decomposition We focus on deterministic algorithms for the Max- imal Independent Set (henceforth, MIS) and coloring problems. These problems are among the most impor- tant problems in symmetry breaking. It has been shown This research has been supported by the Israeli Academy of that it is impossible to break symmetry using determin- Science, grant 483/06. istic algorithms unless each vertex has a distinct iden- ¤ Department of Computer Science, Ben-Gurion University of the tity number [12]. Consequently, whenever deterministic Negev, POB 653, Beer-Sheva 84105, Israel. algorithms for symmetry breaking are concerned, it is E-mail: fleonidba,[email protected] assumed that each vertex has a distinct identity num- 2 ber (henceforth, ID) represented by bit strings of length for any ² > 0). (See Section 2 for a more detailed com- O(log n) [5,10,18,17]. parison between various graph families.) To our knowledge, prior to our work the only graph families on which there existed a sublogarithmic time algorithm for the MIS problem were the family of graphs 1.2 MIS with bounded degree [11,17,18] and the family of graphs with bounded growth [9,14,23]. In other words, our al- A subset I ⊆ V of vertices is called a Maximal Inde- gorithm is the ¯rst sublogarithmic time (deterministic pendent Set (henceforth, MIS) of G if or randomized) algorithm for the MIS problem on any (1) I is an independent set, i.e., for every pair u; w 2 U graph family other than these two families of graphs. of neighbors, either u or w do not belong to I, and Even for the family of unoriented trees, which is con- (2) for every vertex v 2 V , either v 2 I or there exists tained in the family of graphs of constant arboricity, a neighbor w 2 V of v that belongs to I. the best previous result has running time of O(log n). The problem of computing MIS is one of the most In addition, we show that an MIS on graphs of ar- fundamental problems in the area of Distributed Algo- boricityp at most a can be computed deterministically rithms. More than twenty years ago Luby [19] and Alon, in O(a log n + a log a) time. In particular, this result Babai, and Itai [1] devised two logarithmic time ran- implies that an MIS can be computed deterministically domized algorithms for this problem on general graphs. in polylogarithmic time on graphs G with arboricity These algorithms remain the state-of-the-art to this at most polylogarithmic in n. Hence we signi¯cantly date. Awerbuch, Goldberg, Luby, and Plotkin [3] de- extend the class of graphs on which an e±cient (that vised the ¯rst deterministic algorithm for this problem is, requiring a polylogarithmic time) deterministic al- on general graphs, which was later improved by Pan- gorithm for computing MIS is known. conesi and Srinivasan [21] in 1992. The latter algorithmp is the state-of-the-art. Its running time is 2O( log n). The best-known lowerq bound for the MIS problem on general graphs, ­( log n ), is due to Kuhn, Mosci- log log n 1.3 Coloring broda, and Wattenhofer [15]. Cole and Vishkin [5] presented an algorithm for com- We also study the coloring problem. This problem is puting MIS on rings and oriented trees. The running closely related to the MIS problem, and similarly to time of the algorithm of [5] is O(log¤ n). Linial [18] has the latter problem, the coloring problem is one of the shown that this result is tight up to constant factors. most central and most intensively studied problems in In 1988 Goldberg, Plotkin, and Shannon [11] initiated Distributed Algorithms [10,11,18,17,25]. The goal of the study of the MIS problem on sparse graphs. They the coloring problem is to assign colors to vertices so devised a deterministic algorithm for the MIS problem that for each edge e, the endpoints of e are assigned on planar graphs that requires O(log n) time. Their al- distinct colors. In other words, the vertex set has to be gorithm extends also to graphs of bounded genus. partitioned into color classes, such that each color class We improve and generalize the result of Goldberg et forms an independent set. There is an inherent tradeo® al. [11] and devise a deterministic algorithm for the MIS between the running time of a distributed coloring algo- problem on graphs of bounded arboricity that requires rithm and the number of colors it employs for coloring log n the underlying network. time O( log log n ). The arboricity of a graph is a mea- sure for its sparsity. (The de¯nition of arboricity can There are e±cient algorithms for coloring graphs be found in Section 2.) Sparse graphs have low arboric- of bounded degree. Speci¯cally, for a positive integer ity. The family of graphs of bounded arboricity includes parameter ¢, Goldberg, Plotkin, and Shannon [11] de- not only planar graphs, graphs of bounded genus, and vised a (¢ + 1)-coloring algorithm with running time graphs of bounded degree, but also graphs that exclude O(¢ log n). Goldberg and Plotkin [10] devised an O(¢2)- any ¯xed minor and graphs of bounded treewidth. More- coloring algorithm with running time O(log¤ n), for con- over, a graph with constant arboricity may have genus stant values of ¢, and Linial [18] extended this result to p O(n), and may contain K n as a minor. Consequently, general values of ¢. Recently, Kuhn and Wattenhofer the family of graphs on which our algorithm constructs [17] presented a (¢+1)-coloring algorithm with running MIS in sublogarithmic time is much wider than each of time O(¢ log ¢ + log¤ n). For planar graphs, Goldberg the families that we have listed above. Moreover, our et al. [11] devised a 7-coloring algorithm with running result applies also when the arboricity a = a(G) of the time O(log n), and a 5-coloring algorithm with running input graph G is super-constant (up to a = log1=2¡² n, time O(log n log log n). (The latter algorithm assumes 3 that a planar embedding of the input graph is known strate that for a parameter q, q ¸ 1, a forests- decom- to the vertices.) position into O(q ¢ a) forests of a graph with arboricity log n We signi¯cantly extend the class of graphs families a can be computed (distributedly) in time O( log q ). We for which e±cient coloring algorithms are known, and log n ¤ also show a lower bound of ­( log q+log a )¡O(log n) for devise a (b(2 + ²) ¢ ac+1)-coloring algorithm for graphs this problem, demonstrating that our algorithm is near- G of bounded arboricity a(G) · a that has running optimal. Remarkably, all our algorithms can be applied time O(a¢log n). (The parameter ² > 0 can be set as an even when vertices do not know the arboricity of the arbitrarily small positive constant.) In particular, our underlying graph. algorithm 7-colors any graph of arboricity at most 3 in It is plausible that our algorithm for computing logarithmic time, subsuming the result of Goldberg et forests-decompositions will be useful for other applica- al.